722edo

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← 721edo722edo723edo →
Prime factorization 2 × 192
Step size 1.66205¢ 
Fifth 422\722 (701.385¢) (→211\361)
Semitones (A1:m2) 66:56 (109.7¢ : 93.07¢)
Dual sharp fifth 423\722 (703.047¢)
Dual flat fifth 422\722 (701.385¢) (→211\361)
Dual major 2nd 123\722 (204.432¢)
Consistency limit 5
Distinct consistency limit 5

722 equal divisions of the octave (abbreviated 722edo or 722ed2), also called 722-tone equal temperament (722tet) or 722 equal temperament (722et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 722 equal parts of about 1.66 ¢ each. Each step represents a frequency ratio of 21/722, or the 722nd root of 2.

722edo is a strong 2.7.19.23 subgroup tuning, with 179\722 being a semiconvergent to the log2(19/16). Despite having a strong approximation of 7, it is only consistent upwards to the 5-limit.

Using the 421\722 fifth, it supports a variant of fifth-stacked tuning that divides 38th harmonic into 9 parts, meaning that C - D# in this system is equal to 19/16, the otonal minor third. This creates a peculiar violation of Western theory which would require spelling this minor triad involving 19/16 as C-D#-G instead of C-Eb-G. This can be realized as 355 & 722 2.17.19.23 temperament from a regular temperament theory perspective - it should be noted that the fifth is not mapped to 3/2 but is slightly flatter.

Aside from this, 722bc val tempers out the hemifamity comma and is a tuning for the undecental temperament. Since 722 is divisible by 19, the 722dg val is a tuning for the kalium temperament in the 19-limit.

Odd harmonics

Approximation of odd harmonics in 722edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.570 -0.718 +0.149 +0.522 +0.483 +0.470 +0.374 -0.246 -0.006 -0.421 -0.019
Relative (%) -34.3 -43.2 +9.0 +31.4 +29.0 +28.3 +22.5 -14.8 -0.4 -25.3 -1.2
Steps
(reduced)
1144
(422)
1676
(232)
2027
(583)
2289
(123)
2498
(332)
2672
(506)
2821
(655)
2951
(63)
3067
(179)
3171
(283)
3266
(378)

Subsets and supersets

722edo has subset edos 1, 2, 19, 38, 361.